Singular boundary elements for the analysis of cracks in plane strain

Straight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip, and subdominant modes in which it is proportional to r^1·5. The singular functions extend over many boundary elements on each crack face. A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.     

Symmetric Galerkin boundary integral fracture analysis for plane orthotropic elasticity

This paper discusses the formulation and implementation of the symmetric Galerkin boundary integral method for two dimensional linear elastic orthotropic fracture analysis. For the usual case of a traction-free crack, the symmetry of the coefficient matrix can be effectively exploited to significantly reduce the computational work required to construct the linear system. In addition, computation time is reduced by employing efficient analytic integration formulas for the analysis of the orthotropic singular and hypersingular integrals. Preliminary test calculations indicate that the method is both accurate and efficient.     

Complex variable-based analysis for two semi-permeable collinear cracks in a piezoelectromagnetic media

The problem of two equal collinear cracks weakening a poled transversely isotropic piezo-electro-magnetic plate is addressed under semi-permeable electro-magnetic boundary conditions on the crack faces. The problem is formulated employing Stroh formalism and solved using a complex variable technique. Closed form analytical expressions are derived for various fracture parameters. An illustrative numerical case study is presented for poled BaTiO₃ ? CoFe₂O₄ ceramic cracked plate to study the effect of prescribed electric load, magnetic load, and inter-crack distance on fracture parameters. Moreover, a comparative study is done of volume fraction, impermeable, and semi-permeable boundary conditions on fracture parameters.     

A symmetric boundary integral approach to transient poroelastic analysis

 The problem of the transient quasi-static analysis of a poroelastic body subjected to a history of external actions is formulated in terms of four boundary integral equations, using time-dependent Green's functions of the “free” poroelastic space. Some of these Green's functions, not available in the literature are derived “ad hoc”. The boundary integral operator constructed is shown to be symmetric with respect to a time-convolutive bilinear form so that the boundary solution is characterized by a variational property and its approximation preserving symmetry can be achieved by a Galerkin boundary element procedure. Communicated by S. N. Atluri, 1 July 1996     

A study on influence of poling direction on piezoelectric plate weakened by two collinear semi-permeable cracks

A study on the influence of change in poling direction is carried out for mechanical and electric strip-yield model for a transversely isotropic piezoelectric plate cut along two equal collinear semi-permeable straight hairline cracks. Two cases are considered, when the developed saturation zone is bigger or smaller than mechanical yield zone. The Stroh formalism and complex variable technique are employed to obtain the solution. The effect of change in poling direction is studied on crack opening displacement, crack opening potential drop and energy release rate. Also, a study is carried out on the suitability of crack closure for different piezoelectric ceramics. The obtained results are presented graphically and discussed.     

A new boundary integral equation for notch problem of plane elasticity

A new boundary integral equation for the notch problem of plane elasticity is formulated in this paper. In the formulation, the distributed dislocation density is taken to be the unknown function and the resultant force function to be the right-hand term in the resulting integral equation. As a result the integral equation derived contains a logarithmic kernel. The equation is compact in form and convenient for computation. The accuracy of the method is demonstrated through a number of numerical examples.     

The charge-free zone model for electrically conductive cracks in dielectric and piezoelectric ceramics

The charge-free zone model for electrically conductive cracks in dielectric and piezoelectric ceramics is reviewed with the simplified constitutive equations and the regular constitutive equations and then applied to interpret the experimental results on window ceramic glass and poled lead zirconate titanate PIC 151 ceramics. A good agreement is found between the experimental results and the theoretical predictions from the charge-free zone model, which indicates that the concepts of fracture mechanics can be applied to the failure of conductive cracks in dielectric and piezoelectric ceramics.     

Axisymmetric boundary integral equation analysis of interface cracks between dissimilar materials

The numerical boundary integral equation (BIE) method with quadratic quarter-point crack-tip singular elements is used to analyse interface cracks between dissimilar material in axisymmetry. Such crack problems present modelling difficulties using conventional procedures for obtaining the stress intensity factors. This is because of the oscillatorily singular nature of the stresses in the vicinity of the bimaterial interface crack-tip. Analytical expressions for the direct evaluation of the fracture characterising parameters from the BIE numerical results of displacements or tractions are derived. Three different crack problems are investigated, two of which have known solutions in the literature. Excellent agreement between the BIE results and these other established solutions are obtained even with relatively coarse mesh discretisations. The present study illustrates the ease with which the BIE method may be used in the fracture analysis of both straight and curved binaterial interface cracks.     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

Hypersingular integral equations for a thermoelastic problem of multiple planar cracks in an anisotropic medium

The problem of calculating the thermoelastic stress around an arbitrary number of arbitrarily located planar cracks in an infinite anisotropic medium is considered. The cracks open up under the action of suitably prescribed heat flux and traction. With the aid of suitable integral solutions, we reduce the problem to solving a system of Hadamard finite-part (hypersingular) integral equations. The hypersingular integral equations are solved for specific cases of the problem.     

A boundary element method for transient piezoelectric analysis

In this paper, a boundary element (BE) formulation is developed originally which treats three-dimensional problems of transient piezoelectricity. The approach at hand uses the fundamental solution of the static piezoelectric operator instead of the transient one. This results in a domain integral appearing in the representation formula, which contains the inertia term. This domain integral can be transformed to the boundary using the dual reciprocity method (DRM), which leads to a system of ordinary differential equations in time domain, similar to the systems obtained in standard finite element methods (FEM). The DRM has been chosen because of the difficulties and big computational effort involved in a BE implementation, which makes use of the anisotropic transient piezoelectric fundamental solution. It is an approach that appears to be much too time-consuming for use in a commercial BE code, in which computational costs is an important issue. The method presented in this paper is validated by a numerical example for transient piezoelectricy, which demonstrates excellent agreement with FE computations for the generalized displacements, and an improved accuracy for the flux quantities such as electric field and elastic stresses.     

A pure boundary element method approach for solving hypersingular boundary integral equations

This paper is concerned with discretization and numerical solution of a regularized version of the hypersingular boundary integral equation (HBIE) for the two-dimensional Laplace equation. This HBIE contains the primary unknown, as well as its gradient, on the boundary of a body. Traditionally, this equation has been solved by combining the boundary element method (BEM) together with tangential differentiation of the interpolated primary variable on the boundary. The present paper avoids this tangential differentiation. Instead, a “pure” BEM method is proposed for solving this class of problems. Dirichlet, Neumann and mixed problems are addressed in this paper, and some numerical examples are included in it.     

A boundary integral equation approach for the calculation of gradients and higher order derivatives in solid mechanics problems

An asymptotically convergent boundary integral technique is presented for the analytical calculation of the gradients in the solid mechanics problems. Upon introducing the perturbed trial function for the boundary unknowns into the differentiated integral equations, it is proved that the former intractable hyper-singular integral equations turn out to become asymptotically convergent. The current approach is able not only to calculate the gradients at the point close to or just at the boundary, but also to obtain the higher order derivatives conveniently. Compared with other suggested techniques, the current one has the advantages of efficiency and generality. The demonstrative numerical examples show that the results have very good accuracy compared with the close form solution. Besides, the problem of properly selection of the shape functions to represent the deflection along the free or symmetric side in the Kirchhoff plate bending problems is also discussed.The study is supported by the National Natural Science Foundation with project No. 19072010     

A boundary integral equation method in the frequency domain for cracks under transient loading

This paper concerns the fracture mechanics problem for elastic cracked materials under transient dynamic loading. The problem is solved by use of the boundary integral equations in the frequency domain, and the components of the solution are presented by the Fourier exponential series. The dynamic stress intensity factors are computed for different stress pulses and compared with those obtained for the case of the Heaviside loading.     

Multiple interacting planar cracks in an anisotropic multilayered medium under an antiplane shear stress: a hypersingular integral approach

The problem of determining the stress field around an arbitrary number of arbitrarily-located planar cracks in an anisotropic elastic half-space which adheres perfectly to an infinitely-long elastic strip is considered. The strip is made up of several layers of anisotropic materials which are perfectly bonded to one another. The multilayered medium is assumed to undergo an antiplane deformation. Suitable integral expressions are used to represent the displacement and the stress, leading to a system of hypersingular integral equations to be solved. For a specific example of the problem, which involves particular transversely-isotropic materials, the hypersingular integral equations are solved numerically, in order to calculate the relevant crack tip stress intensity factors.     

A boundary integral solution for the problem of multiple interacting cracks in an elastic material

A boundary integral equation method for the solution of a class of two-dimensional elasticity problems involving multiple interacting cracks in an elastic material is presented. The method is used to obtain a procedure for the numerical evaluation of the crack tip stress intensity factors for this class of problems. The stress intensity factors for some specific problems are computed using this procedure.

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Résumé On présente une méthode par équations intégrales de contour pour solutionner une classe de problèmes d'élasticité bidimensionnelle comportant de multiples fissures interactives dans un matériau élastique. On utilise la méthode pour obtenir une procédure d'évaluation numérique des facteurs d'intensité de contraintes à l'extrémité de la fissure, pour cette classe de problèmes. En utilisant cette procédure, on peut calculer les facteurs d'intensité de contraintes correspondant à divers problèmes spécifiques.

     

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.     

Frictional contact problems of kinked cracks modelled by a boundary integral method

A numerical method is presented for the solution of two dimensional crack problems including the effects of crack kinks and frictional contact between crack faces. The metod is based on an integral equation for the resultant forces along a crack. Coulomb friction between contacting crack surfaces is taken into account. The numerical implementation is demonstrated by considerations of surface and sub-surface piece-wise straight line cracks in a half-plane. Numerical results are presented to illustrate the efficiency and the reliability of the presented method.